You are absolutely correct, There is no need to SUPPLY STEAM TO the HP turbine seals once there is pressure inside the HP casing.
Even more so, once the HP casing pressure is above about 1/3 to 1/2 rated, there is no way to supply steam since the pressure trying to escape from the ends will be greater than the sealing pressure. Once the turbine is loading, the steam that would be forced out of the ends of the HP turbine will now flow INTO the steam seal header and the startup steam seal feed valve needs to be closed. But instead of letting the steam blow out of the ends of the HP turbine, that steam can now be used to supply the steam to the LP turbine seals.
so for a design purpose discription, the HP seals are connected to the steam seal header that also provides steam to the LP turbine during startup and low load.
Once at load, the HP seals are connected to the steam seal header so that the waste steam that would blow from the ends can be used to seal the LP turbine and the original main steam used for sealing during startup is no longer needed and used to make MWs. So you are also correct in that there is another function
Given that the design has an uncontrollable supply of sealing steam from the HP turbine at load, the steam seal header MUST have a means to control the pressure to the LP seals. the steam seal controls will close the feed as the pressure increases and then open a DUMP valve to releive the excessive flow from the HP turbine seals. this dump can go directly to the condensor (and thus be wasted) or it can be routed to a low pressure heater.
the link in my first post shows a control scheme for the steam seal header
the startup feed valve - SSFV
the at load dump valve - SPUV